Question: Simplify and expand the following expression: $ \dfrac{y + 5}{4y + 3}+\dfrac{2y}{2y - 5} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4y + 3)(2y - 5)$ Multiply the first term by $\dfrac{2y - 5}{2y - 5}$ $ \begin{align*} \dfrac{y + 5}{4y + 3} \times \dfrac{2y - 5}{2y - 5} & = \dfrac{(y + 5)(2y - 5)}{(4y + 3)(2y - 5)} \\ & = \dfrac{2y^2 + 5y - 25}{(4y + 3)(2y - 5)}\end{align*} $ Multiply the second term by $\dfrac{4y + 3}{4y + 3}$ $ \begin{align*} \dfrac{2y}{2y - 5} \times \dfrac{4y + 3}{4y + 3} & = \dfrac{(2y)(4y + 3)}{(2y - 5)(4y + 3)} \\ & = \dfrac{8y^2 + 6y}{(2y - 5)(4y + 3)}\end{align*} $ Now we have: $ = \dfrac{2y^2 + 5y - 25}{(4y + 3)(2y - 5)} + \dfrac{8y^2 + 6y}{(2y - 5)(4y + 3)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{2y^2 + 5y - 25 + 8y^2 + 6y}{(4y + 3)(2y - 5)} $ $ = \dfrac{10y^2 + 11y - 25}{(4y + 3)(2y - 5)}$ Expand the denominator: $ = \dfrac{10y^2 + 11y - 25}{8y^2 - 14y - 15}$